Game Analysis
Juan Rocha
2 May 2016
The game
Now I turn your attention to the game data. On groups of 4 we played with the 256 fishermen a fishing game, where each person were supposed to make a decision on how much to fish. Each group of 4 fishermen shared a common fish stock that started with 50 fish. Their decisions were private but they could communicate throghout the game. The game lasted 16 rounds unless they collapsed their resource, of which 6 rounds where for learning the game (‘base line within treatments’) and another 10 rounds they had to face one of four possible situations or treatments. Base line groups continued playing the same game as the first 6 rounds. Threshold treatment players were informed that a climate event has occurred decreasing the reproduction rate of the resource. Risk treatment players faced a similar situation but the event was not certain, they knew that the probability of the event happening was of 50% chance. Uncertainty treatment also faced the possibility of a climate event on the second part of the game but players didn’t know the actual probability of the event. The figure below shows a summary of all time lines of the game per place and per treatment. The blue line shows the mean of the time series while the shaded area shows the smooth, this is the Gaussian confidence intervals based on the t-distribution (so we assume normality). The red line marks round 6, this is the end of the first part of the game. It helps to caution the reader that in round 7 the treatment started and we reset the fish stock size to 50 as in the beginning of the game, that’s why in all timelines there is a ‘jump’ from round 6 to 7.
dat <- read.csv(file="~/Dropbox/BEST/Colombia/0_Game data/160427_corrected_full_data_long.csv", row.names=1)
g <- ggplot(dat=dat, aes(x=Round, y=value)) +
geom_vline( aes( xintercept=6, color='red', alpha=0.1), show.legend = F) +
stat_summary(fun.data='mean_cl_normal', geom='smooth') + # option 'mean_cl_boot I like the most but normal assumes normality
facet_grid(Treatment ~ Place)
g
Now imagine each individual is exploring the parameter space of the stock size at the beginning of each round (which depends on what the group did on the previous round), and her/his own decision. Each individual will then follow a trajectory on that phase diagram that potentially capture a signature of her/his strategy. The figure below shows such trajectory, where each line represents the time series of each person for the afternoon session in Las Flores on February 4th. We played simultaneously with 16 players, 4 in each of the 4 above mentioned treatments. Here we see that the base line group collapsed twice, once on round 5 and another in round 11; this is why we see the line broken and never reaching the light blue colour that denotes the end of the game. Here it also comes clear that, for example, people who played the threshold treatment collectively chose to stay on high levels of the resource by fishing maximum 4 units and maintaining their resource above 38 fishes.
datP1 <- dat %>%
filter(dat$Place == 'Las Flores' & dat$Session == 'pm' & Date == "2016-02-04" ) %>%
gdata::drop.levels()
p <- ggplot(data=datP1, aes(x=value, y=StockSizeBegining), group=Player) +
geom_path(aes(color=Round)) + facet_grid(Treatment~Player) +
ggtitle('Phase space trajectory per player\n Las Flores, 2016-02-04-pm ') +
theme(text= element_text(family='Helvetica', size=9))
p
At the group level, the same rationale can be applied if one plots the stock size at the begining of each round against the sum of the total catch for all 4 participants per group. Note that we have in total 64 groups (4 treatments * 4 sessions * 4 locations). The figure below summarizes all 64 groups, each one represented with a colour, and each colour showing the contour or the parameter space where most points were concentrated by location and treatment. The contour was calculated with a 2-dimensional smoother (bandwidth = 15). The figure on the left shows the aggregated group behaviour by ploting the sum of the total catch. The figure on the right shows the resource dynamics at the beginning of each round vs. the resource at the end of the round which is the same as the new stock size for the next round, this is what people fish plus the reproduction in the round. In dynamical systems this is equivalent to \(f(x[t+1]):x[t]\).
c <- ggplot(data=dat, aes(y=StockSizeBegining, x=SumTotalCatch), group=Session) +
stat_density_2d(aes(color=group, alpha=0.5), n=100, h=15, show.legend = F) +
facet_grid( Treatment ~ Place)+
theme(text= element_text(family='Helvetica', size=9))
p <- ggplot(data=dat, aes(y=NewStockSize, x=StockSizeBegining), group=Session) +
geom_abline(aes(color='black', alpha=0.5),slope=1, intercept = 0) +
stat_density_2d(aes(color=group, alpha=0.3), n=100, h=15, show.legend = F) +
geom_point(aes(color=group, alpha=0.3), show.legend=F) +
facet_grid( Treatment ~ Place)+
theme(text= element_text(family='Helvetica', size=9))
grid.arrange ( c,p, ncol=2, nrow=1 )
The next natural question to ask is whether there is an effect between:
- The first and the second part of the game
- Treatments
- Place
The four panels below try to get to these differences
pm1 <- ggpairs(data= filter(dat, dat$part == F),
columns=c('StockSizeBegining', 'NewStockSize', 'Treatment','Place'),
upper= list(continuous='density'), lower=list(continuous='points'),
mapping=aes(color= Place, alpha=0.5), title='Color by place, first part')
pm2 <- ggpairs(data= filter(dat, dat$part == T),
columns=c('StockSizeBegining', 'NewStockSize', 'Treatment','Place'),
upper= list(continuous='density'), lower=list(continuous='points'),
mapping=aes(color= Place, alpha=0.5), title='Color by place, second part')
pm3 <- ggpairs(data= filter(dat, dat$part == F),
columns=c('StockSizeBegining', 'NewStockSize', 'Treatment','Place'),
upper= list(continuous='density'), lower=list(continuous='points'),
mapping=aes(color= Treatment, alpha=0.5), title='Color by treatment, second part')
pm4 <- ggpairs(data= filter(dat, dat$part == T),
columns=c('StockSizeBegining', 'NewStockSize', 'Treatment','Place'),
upper= list(continuous='density'), lower=list(continuous='points'),
mapping=aes(color= Treatment, alpha=0.5), title='Color by treatment, second part')
#grid.arrange(
pm1 + theme(text= element_text(family='Helvetica', size=6))
pm2 + theme(text= element_text(family='Helvetica', size=6))
pm3 + theme(text= element_text(family='Helvetica', size=6))
pm4 + theme(text= element_text(family='Helvetica', size=6))
# ncol=2, nrow=2
#)
Fishing styles
Something I’d really like to get into (for modeling purposes) is identifying fishing strategies or styles. To do so I ran a PCA and MDS on the time series (16 rounds) of each of the 256 players in our dataset. The results are shown below.
library(vegan)
players <- (reshape::cast(dat, ID_player ~ Round))[,-1] #delete playersID
place <- reshape::cast(dat, ID_player ~ Place)[,-1] # 16 because calculate length of 'value'
group <- reshape::cast(dat, ID_player ~ group)[,-1]
treat <- reshape::cast(dat, ID_player ~ Treatment)[,-1]
context <- cbind(place,treat) # don't use group yet, maybe for aes
pca <- rda(players, context) #, context
# note: I ran mds with 'euclidean' distance and 'manhattan'. Euclidean gave an error saying there is not enough data. manhattan calculated but didn't reach convergence. Morisita, jaccard, kulczynski, horn, rau, don't reach convergence neither. Frequent error on different measures: Stress is (nearly) zero - you may have insufficient data. It does work however if I use the transpose. It calculates the ordering of rounds on mds$points, but the ordering of players can be found too at mds$species
mds <- metaMDS(players, dist= 'manhattan', trymax=400, autotransform=F, k=2)
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## ... procrustes: rmse 0.00579523 max resid 0.06538604
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## ... procrustes: rmse 0.008867218 max resid 0.1013378
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## ... procrustes: rmse 0.005746816 max resid 0.06480613
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## ... procrustes: rmse 0.001836863 max resid 0.0211627
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## ... procrustes: rmse 0.006550399 max resid 0.07906555
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## ... procrustes: rmse 0.009063853 max resid 0.1016625
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## ... procrustes: rmse 0.006691132 max resid 0.1006848
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## ... procrustes: rmse 0.004206352 max resid 0.0640644
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## ... procrustes: rmse 0.009387307 max resid 0.1023246
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## ... procrustes: rmse 0.006663923 max resid 0.07925868
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## ... procrustes: rmse 0.008449541 max resid 0.101199
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## ... procrustes: rmse 0.006224877 max resid 0.06695016
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## ... procrustes: rmse 0.006587365 max resid 0.07794172
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# mds2 <- metaMDS(players, dist= 'mahalanobis', trymax=1000, autotransform=F, k=3, previous.best = mds) # solution from http://stackoverflow.com/questions/14434794/no-stable-solution-using-metamds-in-vegan
ef1 <- envfit(mds, context, permu=999)
par(mfrow=c(1,3))
plot(pca)
plot(mds, type='p', display=c('sites', 'species'), cex=0.8)
plot(ef1, p.max=0.05, col='blue', cex=0.8)
stressplot(mds)
PCA always gives some results (but meaningful?) and MDS very often runs into problems of non-convergence. PCA and MDS assumes that observations are independent. Here, they aren’t because what a player does influence what other players do, so they are not independent among group members, but they are between groups. In addition, since in the game what happens in the future depends of what people played in the past, there is not independence between rounds neither. After a long talk with Ingo, a possible solution is running the ordination method at the group level, after de-trending the data with e.g. demean it.
dat <- mutate(dat, val.dtrend = value - mean(value))
players <- mutate(players, avg.log = apply(players, 1, function (x) {log10(mean(x))} ),
sd.log = apply (players, 1, function (x) {log10(sd(x))})
)
p1 <- ggplot(players, aes(x=avg.log, y=sd.log)) +
geom_abline(intercept=0, slope=0.55) +
geom_point() #+ scale_x_log10() + scale_y_log10()
p1
# The lines below should produce the same plot but is not working
# p1 <- ggplot(dat, aes(x=mean(value), y=sd(value), group = levels(ID_player))) +
# # geom_abline(intercept=0, slope=0.55) +
# geom_point( show.legend = F) #+ scale_x_log10() + scale_y_log10()
To-Do
- ggpairs plots for treatments, place, and before/after
- statistics: t-test, Mann-Witness test, Fisher exact test for all
- some simple regressions
- styles clustering
g <- ggplot(dat, aes(x=value, y=StockSizeBegining,color=Treatment )) +
geom_point() +
geom_smooth( method='loess') +
ggtitle('Local polynomial regression\n Stock size vs Decision')
g